Decoding method and apparatus

ABSTRACT

A method of decoding is provided comprising processing iterations. In each processing iteration, there is a first Max-Log-MAP decoding operation giving rise to a systematic error due to the Max-Log approximation, and a first weighting operation to weight extrinsic information from the first decoding operation to be applied as a priori information to the second Max-Log-MAP decoding operation. This is followed by a second Max-Log-MAP decoding operation, also giving rise to a systematic error due to the Max-Log approximation, and a second weighting operation to weight extrinsic information from the second decoding to be applied as a priori information to the first Max-Log-MAP decoding of the next iteration. The weights are applied to compensate for the systematic error due to the Max-Log approximation made in the last Max-Log-MAP decoding operation.

FIELD OF THE INVENTION

The present invention relates to decoding, and more particularly, todecoding using a Max-Log-MAP decoding scheme.

DESCRIPTION OF THE RELATED ART

In the field of wireless telecommunications, in particular code divisionmultiple access (CDMA), the demand for low-cost and low-power decoderchips, particularly for use in mobile user terminals has resulted inrenewed interest in low-complexity decoders.

Several known approaches seeking to reduce complexity of an optimumMaximum A posteriori Probability (MAP) decoder are known, such as theLog-MAP and Max-Log-MAP schemes.

SUMMARY OF THE INVENTION

A method of decoding and a decoding apparatus according to the presentinvention are defined in the independent claims to which the readershould now refer. Preferred features are laid out in the dependentclaims.

An example of the present invention is a method of decoding comprisingprocessing iterations. In each processing iteration, there is a firstMax-Log-MAP decoding operation giving rise to a systematic error due tothe Max-Log approximation, and a first weighting operation to weightextrinsic information from the first decoding operation to be applied asa priori information to the second Max-Log-MAP decoding operation. Thisis followed by a second Max-Log-MAP decoding operation, also giving riseto a systematic error due to the Max-Log approximation, and a secondweighting operation to weight extrinsic information from the seconddecoding to be applied as a priori information to the first Max-Log-MAPdecoding of the next iteration. The weights are applied to compensatefor the systematic error due to the Max-Log approximation made in thelast Max-Log-MAP decoding operation.

It can thus be considered that a modification to a known Max-Log-MAPiterative decoder is provided, basically using correction weights forthe extrinsic information at each iteration in order to correct theerror caused by the Max-Log approximation in the extrinsic informationprovided by the previous decoding iteration. This can be achieved byapplying optimised weight factors to the extrinsic information in eachdecoding iteration. Applying such weights not only allows the inherentadvantages of a Max-Log-MAP decoder to be kept, such as of lowcomplexity and insensitivity to input scaling, but tends to result inimproved performance.

BRIEF DESCRIPTION OF THE DRAWINGS

An embodiment of the present invention will now be described by way ofexample and with reference to the drawings, in which:

FIG. 1 is a diagram illustrating a turbo-decoder for parallelconcatenated codes (PRIOR ART),

FIG. 2 is a diagram illustrating a known Max-Log-MAP decoder (PRIORART),

FIG. 3 is a diagram illustrating a Max-Log-MAP decoder according to thepresent invention,

FIG. 4 is a diagram illustrating determination of weights to be appliedto the Max-Log-MAP decoder shown in FIG. 3, and

FIG. 5 is a diagram illustrating a turbo-decoder for parallelconcatenated codes including the Max-Log-MAP decoder shown in FIGS. 3and 4.

DETAILED DESCRIPTION

As background, turbo-coding will first be explained generally, beforefocussing in on Log-MAP decoding and then Max-Log-MAP decoding. Animprovement to Max-Log-MAP decoding will then be presented.

Turbo-Decoding

FIG. 1 shows a known turbo-decoder 2 for a parallel concatenated code ofrate ⅓ (i.e. where one information bit has two associated parity bits),as presented for example in the book by B. Vucetic and J.Yuan, entitled“Turbo codes”, published by Kluwer Academic Publishers, 2000. The codeis termed parallel concatenated because two known recursive systematicconvolutional encoders (not shown) within a turbo-encoder (not shown)have operated on the same set of input bits rather than one encoding theoutput of the other.

Given systematic (i.e. information) bit x_(t,0) and parity (i.e. checksequence) bits x_(t,1) and x_(t,2), generated at the turbo-encoder (notshown) and assuming transmission through an additive white gaussiannoise (AWGN) channel at time t, the corresponding received signals atthe turbo-decoder 2 may be written as Λ_(c)(x_(t,0)), Λ_(c)(x_(t,1)) andΛ_(c)(x_(t,2)). Turbo decoding is performed in an iterative manner usingtwo soft-output decoders 4,6, with the objective of improving dataestimates from iteration i to iteration i+1. Each soft-output decoder4,6 generates extrinsic information Λ_(e) ^(i)(x_(t)) on the systematicbits which then serves as a priori information Λ_(a) ^(i)(x_(t,0)) forthe other decoder 6,4. Extrinsic information is the probabilisticinformation gained on the reliability of the systematic bits. Thisinformation is improved on through decoding iterations. In order tominimise the probability of error propagation, the decoders 4, 6 areseparated by interleaving process such that extrinsic information bitspassing from decoder 4 to decoder 6 are interleaved, and extrinsicinformation bits passing from decoder 6 to decoder 4 are de-interleaved.

As regards, a choice of soft output decoders, 4,6, a maximum aposteriori probability (MAP) scheme would be the optimum decoding schemein the sense that it results in a minimum probability of bit error.However, the MAP scheme is computationally complex and, as a result isusually implemented in the logarithmic domain in the form of Log-MAP orMax-Log-MAP scheme. While the former is a mathematical equivalent ofMAP, the latter scheme involves an approximation which results in evenlower complexity, albeit at the expense of some degradation inperformance.

For further background, the reader is referred to the book by B. Vuceticand J. Yuan, entitled “Turbo codes”, published by Kluwer AcademicPublishers, 2000.

Log-MAP Algorithm

The known log-domain implementation of the MAP scheme requireslog-likelihood ratios (LLR) of the transmitted bits at the input of thedecoder. These LLRs are of the form $\begin{matrix}{{\Lambda(x)} = {{\log\quad\frac{\Pr\left( {x = {+ 1}} \right)}{\Pr\left( {x = {- 1}} \right)}} = {\frac{2}{\sigma_{n}^{2}}r}}} & (1)\end{matrix}$where Pr(A) represents the probability of event A, x is the value of thetransmitted bit, r=x+n is the received signal at the output of anadditive white gaussian noise (AWGN) channel where x is the data value,and n is the noise assumed to have an expected value E{|n|²}=σ_(n) ²where σ_(n) ² is the variance of the noise.

Given LLRs for the systematic and parity bits as well as a priori LLRsfor the systematic bits, the Log-MAP decoder computes new LLRs for thesystematic bits as follows: $\begin{matrix}\begin{matrix}{{\Lambda\left( x_{t,0} \right)} = {\log\frac{\sum\limits_{t = 0}^{M_{s} - 1}{\exp\left\{ {{{\overset{\_}{\alpha}}_{t - 1}\left( l^{\prime} \right)} + {{\overset{\_}{\gamma}}_{t}^{1}\left( {l^{\prime},l} \right)} + {{\overset{\_}{\beta}}_{t}(l)}} \right\}}}{\sum\limits_{t = 0}^{M_{s} - 1}{\exp\left\{ {{{\overset{\_}{\alpha}}_{t - 1}\left( l^{\prime} \right)} + {{\overset{\_}{\gamma}}_{t}^{0}\left( {l^{\prime},l} \right)} + {{\overset{\_}{\beta}}_{t}(l)}} \right\}}}}} \\{= {{\Lambda_{a}\left( x_{t} \right)} + {\Lambda_{c}\left( x_{t} \right)} + {\Lambda_{e}\left( x_{t} \right)}}}\end{matrix} & (2)\end{matrix}$where γ_(t) ^(q)(l′,l) denotes the logarithmic transition probabilityfor a transition from state l′ to state l of the encoder trellis at timeinstant t given that the systematic bit takes on value q{1,0} and M_(s)is the total number of states in the trellis. (For further explanationof trellis structures the reader is again referred to the Vucetic andYuan book).

Note that the new information at the decoder output regarding thesystematic bits is encapsulated in the extrinsic information termΛ_(e)(x_(t,0)). Coefficients α_(t)(l′) are accumulated measures oftransition probability at time t in the forward direction in thetrellis. Coefficients β_(t)(l) are accumulated measures of transitionprobability at time t in the backward direction in the trellis. For adata block corresponding to systematic bits x_(1,0) to x_(t,0) andparity bits x_(1,1) to x_(t,1), these coefficients are calculated asdescribed below.

Using the following initial values in a forward direction:{overscore (α)}₀(0)=0 and {overscore (α)}₀ (l)=−∞ for l≠0   (4),the coefficients are calculated as $\begin{matrix}{{{{\overset{\_}{\gamma}}_{t}^{q}\left( {l^{\prime},l} \right)} = \frac{{\left\{ {{\Lambda_{a}\left( x_{t,0} \right)} + {\Lambda_{c}\left( x_{t,0} \right)}} \right\} x_{t,0}^{q}} + {{\Lambda_{c}\left( x_{t,1} \right)}x_{t,1}^{q}}}{2}},{and}} & (5) \\{{{\overset{\_}{\alpha}}_{t}(l)} = {\sum\limits_{l^{\prime} = 0}^{M_{s} - 1}{\sum\limits_{q \in {({0,1})}}{\exp{\left\{ {{{\overset{\_}{\alpha}}_{t - 1}\left( l^{\prime} \right)} + {{\overset{\_}{\gamma}}_{t}^{q}\left( {l^{\prime},l} \right)}} \right\}.}}}}} & (6)\end{matrix}$

Using the following initial values in the backward direction:{overscore (β)}_(t)(0)=0 and {overscore (β)}_(t)(l)=−∞ for l≠0   (7),the coefficients are calculated as $\begin{matrix}{{{\overset{\_}{\beta}}_{t}(l)} = {\sum\limits_{l^{\prime} = 0}^{M_{s} - 1}{\sum\limits_{q \in {({0,1})}}{\exp{\left\{ {{{\overset{\_}{\beta}}_{t + 1}\left( l^{\prime} \right)} + {{\overset{\_}{\gamma}}_{t + 1}^{q}\left( {l^{\prime},l} \right)}} \right\}.}}}}} & (8)\end{matrix}$

Equation (2) is readily implemented using the known Jacobian equalitylog(e ^(δ) ¹ +e ^(δ) ² )=max(δ₁, δ₂)+log(1+^(−|δ) ² ^(−δ) ¹ |)   (9)and using a look up table to evaluate the correction functionlog(1+e^(−|δ) ² ^(−δ) ¹ ^(|)).Max-Log-MAP decoding

It is known that the complexity of the Log-MAP scheme can be furtherreduced by using the so-called Max-Log approximation, namelylog(e ^(δ) ¹ +e ^(δ) ² + . . . )≈max(δ₁,δ₂, . . . )   (10)for evaluating Equation (2). (log, of course, denotes natural logarithm,i.e. log_(e)). The Max-Log-MAP scheme is often the preferred choice forimplementing a MAP decoder, for example as shown in FIG. 2, since it hasthe advantage that its operation is insensitive to a scaling of theinput LLRs. This means that knowledge or estimation of the channel'snoise variance σ_(n) ² is not required.

However, the known Max-Log approximation leads to accumulating a bias inthe decoder output (extrinsic information), i.e. the Max-Logapproximation results in biased soft outputs. A bias is, of course, anaverage of errors over time. This is due to the fact that the knownMax-Log-MAP scheme uses the mathematical approximation of Equation (10)to simplify the computation of extrinsic information Λ_(e)(x_(t,0)).This approximation results in an error which accumulates from iterationto iteration and impedes the convergence of the turbo-decoder. Since ina turbo-decoder, each decoder output becomes a priori information forthe following decoding process, the bias leads to sub-optimal combiningproportions between the channel input and the a priori information,thereby degrading the performance of the decoder. In consequence, theknown turbo-decoding process may not converge when the known Max-Log-MAPscheme is used for the constituent decoding processes.

Maximum Mutual Information Combining

The inventors recognised these errors as a problem and so wanted tocorrect for such errors in an efficient manner whilst maintaining thebenefits of the Max-Log-MAP approach.

As mentioned above (see Equation (5)), the central operation in theknown Max-Log-MAP scheme is the computation of logarithmic transitionprobabilities of the form $\begin{matrix}{{{\overset{\_}{\gamma}}_{t}^{q}\left( {l^{\prime},l} \right)} = {\frac{1}{2}\left\{ {{\left\{ {{\Lambda_{a}\left( x_{t,0} \right)} + {\Lambda_{c}\left( x_{t,0} \right)}} \right\} x_{t,0}^{q}} + {{\Lambda_{c}\left( x_{t,1} \right)}x_{t,1}^{q}}} \right\}}} & (14)\end{matrix}$where Λ_(a)(x_(t,0)), Λ_(c)(x_(t,0)) and Λ_(c)(x_(t,1)) are inputs tothe constituent decoder.

The inventors realised that bias produced by the Max-Log-MAP schemeshould be corrected by appropriate scaling of the terms Λ_(a)(x_(t,0))and Λ_(c)(x_(t,0)) in the above equation by weights w_(a) ^(i) and w_(c)^(i), resulting in $\begin{matrix}{{{\overset{\_}{\gamma}}_{t}^{q}\left( {l^{\prime},l} \right)} = {\frac{1}{2}\left\{ {{\left\{ {{w_{a}^{i}{\Lambda_{a}\left( x_{t,0} \right)}} + {w_{c}^{i}{\Lambda_{c}\left( x_{t,0} \right)}}} \right\} x_{t,0}^{q}} + {{\Lambda_{c}\left( x_{t,1} \right)}x_{t,1}^{q}}} \right\}}} & (15)\end{matrix}$where i represents the iteration index. This is illustrated in FIG. 3,which shows a weight w_(a) ^(i) applied to a priori informationΛ_(a)(x_(t,0)), and a weight w_(c) ^(i) applied to systematicinformation (LtRs) Λ_(c)(x_(t,0)) applied to a Max-Log-MAP decoder. Theweights are normalised as described later in this text such that theratio between the LLRs representing systematic and parity bits is notdisturbed.

This correction is simple and effective, and retains the advantage ofMax-Log-MAP decoding in that soft inputs in the form of scaled LLRs areaccepted.

Determination of Weights

The optimum values of the weights w_(a) ^(i) and w_(c) ^(i) to beapplied are those that maximise the transfer of mutual information fromone Max-Log-MAP decoder to the other at every iteration. Mutualinformation is, of course the level of information as to knowledge of adata sequence. These optimum values can be mathematically written as$\begin{matrix}{\begin{bmatrix}w_{{OPT},a}^{i} \\w_{{OPT},c}^{i}\end{bmatrix} = {\left( R_{ɛ}^{i} \right)^{\frac{T}{2}}{eig}_{\max}\left\{ {\left( R_{ɛ}^{i} \right)^{\frac{- 1}{2}}\left( R_{\lambda + ɛ}^{i} \right)^{\frac{T}{2}}\left( R_{ɛ}^{i} \right)^{\frac{- T}{2}}} \right\}}} & (26)\end{matrix}$where eig_(max)(A) denotes the eigenvector corresponding to the largesteigenvalue of matrix A, and R are correlation matrices defined later onin this text. Equation (26) has more than one solution, but any of thesolutions is optimum in maximising mutual information.

The above-mentioned optimum weights are both normalised (divided) byw_(OPT,c) ^(i) so that the weights to be applied become $\begin{matrix}{\begin{bmatrix}w_{{OPT},a}^{i} \\w_{{OPT},c}^{i}\end{bmatrix}->\begin{bmatrix}{w_{{OPT},a}^{i}/w_{{OPT},c}^{i}} \\1\end{bmatrix}} & (27)\end{matrix}$

This normalisation is required to ensure that the natural ratio betweenΛ_(c)(x_(t,0)) and Λ_(c)(x_(t,1)) remains undisturbed.

Computation of the weights in Equation (26) requires the (2×2) matricesR_(ε) ^(i)=R_(λ+ε) ^(i)−R_(λ) ^(i) and R_(λ+ε) ^(i) to be computed.These are: $R_{\lambda + ɛ}^{i} = \begin{bmatrix}{E\left\{ {{\Lambda_{a}^{i}\left( x_{t,0} \right)}{\Lambda_{a}^{i}\left( x_{t,0} \right)}} \right\}} & {E\left\{ {{\Lambda_{a}^{i}\left( x_{t,0} \right)}{\Lambda_{c}^{i}\left( x_{t,0} \right)}} \right\}} \\{E\left\{ {{\Lambda_{c}^{i}\left( x_{t,0} \right)}{\Lambda_{a}^{i}\left( x_{t,0} \right)}} \right\}} & {E\left\{ {{\Lambda_{c}^{i}\left( x_{t,0} \right)}{\Lambda_{c}^{i}\left( x_{t,0} \right)}} \right\}}\end{bmatrix}$ $R_{\lambda}^{i} = \begin{bmatrix}{E\left\{ {{\lambda_{a}^{i}\left( x_{t,0} \right)}{\lambda_{a}^{i}\left( x_{t,0} \right)}} \right\}} & {E\left\{ {{\lambda_{a}^{i}\left( x_{t,0} \right)}{\lambda_{c}^{i}\left( x_{t,0} \right)}} \right\}} \\{E\left\{ {{\lambda_{c}^{i}\left( x_{t,0} \right)}{\lambda_{a}^{i}\left( x_{t,0} \right)}} \right\}} & {E\left\{ {{\lambda_{c}^{i}\left( x_{t,0} \right)}{\lambda_{c}^{i}\left( x_{t,0} \right)}} \right\}}\end{bmatrix}$where Λ_(a) ^(i)(x_(t,0))=λ_(t,a) ^(i)+ε_(t,a) ^(i) and Λ_(c)^(i)(x_(t,0))=λ_(t,c) ^(i)ε_(t,c) ^(i), where λ_(t,a) ^(i) and λ_(t,c)^(i) the notional uncorrupted LLR values, ε_(t,a) ^(i) and ε_(t,c) ^(i)are the errors in LLR as detected compares to the notional uncorruptedLLR values, and “E” denotes statistical mean.

It is proposed that the above statistical means be computed by averagingover a number of data blocks containing a total of say, N, informationbits. In other words: $\begin{matrix}{{R_{\lambda + ɛ}^{i} = \begin{bmatrix}{\frac{1}{N}{\sum\limits_{t = 1}^{N}{{\Lambda_{a}^{i}\left( x_{t,0} \right)}{\Lambda_{a}^{i}\left( x_{t,0} \right)}}}} & \left. {\frac{1}{N}{\sum\limits_{t = 1}^{N}{{\Lambda_{a}^{i}\left( x_{t,0} \right)}{\Lambda_{c}^{i}\left( x_{t,0} \right)}}}} \right\} \\{\frac{1}{N}{\sum\limits_{t = 1}^{N}{{\Lambda_{c}^{i}\left( x_{t,0} \right)}{\Lambda_{a}^{i}\left( x_{t,0} \right)}}}} & {\frac{1}{N}{\sum\limits_{t = 1}^{N}{{\Lambda_{c}^{i}\left( x_{t,0} \right)}{\Lambda_{c}^{i}\left( x_{t,0} \right)}}}}\end{bmatrix}}{and}} & (28) \\{{R_{\lambda}^{i} = \begin{bmatrix}{\alpha^{i}\alpha^{i}} & {\alpha^{i}\beta^{i}} \\{\alpha^{i}\beta^{i}} & {\beta^{i}\beta^{i}}\end{bmatrix}}{where}} & (29) \\{\alpha^{i} = {{\frac{1}{N}{\sum\limits_{t = 1}^{N}{{\Lambda_{a}^{i}\left( x_{t,0} \right)}x_{t,0}\quad\beta^{i}}}} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}{{\Lambda_{c}^{i}\left( x_{t,0} \right)}x_{t,0}}}}}} & (30)\end{matrix}$

The above operations to determine weights are performed only once andoff-line.

The averaging operations defined by equation (30) are undertaken in theaverage determining processor 8 shown in FIG. 4. Weight determinator 10then calculated the weights according to equation (26) and normalises asper equation (27).

Example System

FIG. 5 shows, by way of review, the resulting turbo-decoder 2′ for aparallel concatenated code of rate ⅓ (i.e. where one information bit hastwo associated parity bits).

Given systematic (i.e. information) bit x_(t,0) and parity (i.e. checksequence) bits x_(t,1) and x_(t,2), generated at the turbo-encoder (notshown) and assuming transmission through an additive white gaussiannoise (AWGN) channel at time t, the corresponding received signals atthe turbo-decoder 2′ may be written as Λ_(c)(x_(t,0)), Λ_(c)(x_(t,1))and Λ_(c)(x_(t,2)).

Turbo decoding is performed in an iterative manner using two Max-log-MAPdecoders 4′,6′ of known type as described above, with the objective ofimproving the data estimates from iteration i to iteration i+1. Eachsoft-output decoder 4′,6′ generates extrinsic information Λ_(e)^(i)(x_(t)) on the systematic bits which then serves as a prioriinformation Λ_(a) ^(i)(x_(t,0)) for the other decoder. The Extrinsicinformation is the probabilistic information gained on the reliabilityof the systematic bits. This information is improved on through decodingiterations. In order to minimise the probability of error propagation,the decoders 4′, 6′ are separated by interleaving process such thatextrinsic information bits passing from decoder 4′ to decoder 6′ areinterleaved, and extrinsic information bits passing from decoder 6′ todecoder 4′ are de-interleaved.

Importantly, the a priori information is weighted as described above toremove the bias caused by the Max-log approximation.

The weights are determined by feeding the turbo-decoder with the channeloutputs {Λ_(c)(x_(t,0)), Λ_(c)(x_(t,0)),Λ_(c)(x_(t,1))}t=1 . . . N andperforming I turbo iterations as per normal operation. By observing theconstituent decoder inputs Λ_(c)(x_(t,0)) and Λ_(a)(x_(t,0)) at eachiteration, the coefficients w_(a) ^(i) are computed following Equations(26) to (30) as explained above.

To review, it was found that the performance of a decoder can beimproved by using the modified Max-Log-MAP decoder to approach that of adecoder using the optimum Log-MAP or MAP decoders. This is achieved atthe expense of only two additional multiplications (each multiplicationbeing to apply a weight to a priori information for each decoder) periteration for each systematic bit. The weights are to correct for a biascaused by the Max-Log approximation. The advantages of a Max-Log-MAPdecoder can be maintained, namely its insensitivity to scaling of thelog-likelihood, and that an estimate of noise variance is not required.

Since the weights need only be computed once for a particularturbo-decoder, the improved decoder retains the low complexity of aMax-Log-MAP approach. The values of optimal weights to be applied can becomputed off-line.

1. A method of decoding comprising processing iterations, eachprocessing iteration comprising: performing a first Max-Log-MAP decodingoperation giving rise to a systematic error due to the Max-Logapproximation, performing a first weighting operation of applyingweights to extrinsic information from the first decoding operation to beapplied as a priori information to the second Max-Log-MAP decodingoperation, performing a second Max-Log-MAP decoding operation givingrise to a systematic error due to the Max-Log approximation, andperforming a second weighting operation of applying weights to extrinsicinformation from the second decoding to be applied as a prioriinformation to the first Max-Log-MAP decoding of the next iteration, inwhich each weighting operation is performed to compensate for thesystematic error due to the Max-Log approximation made in the previousMax-Log-MAP decoding operation.
 2. A method according to claim 1, inwhich, of the weights, a weight (w_(a) ^(i)) is applied to a prioriinformation (Λ_(a)(x_(t,0)))which is in the form of a Log LikelihoodRatio about a parity bit, and a weight (w_(c) ^(i)) is applied tofurther a priori information (Λ_(c)(x_(t,0))) which is in the form of aLog Likelihood Ratio about a systematic bit.
 3. A method according toclaim 1, in which values of the weights (w_(a) ^(i), w_(c) ^(i)) to beapplied relate to optimum weights, the optimum weights being optimum inmaximising the transfer of mutual information from one Max-log-MAPdecoding operation to the next.
 4. A method according to claim 3, inwhich the optimum weights are normalised to provide the weights to beapplied so as to maintain a natural ratio between Λ_(c)(x_(t,0)) andΛ_(c)(x_(t,1)) undisturbed.
 5. A method according to claim 1, in whichthe weights to be applied are computed once and used repeatedlythereafter.
 6. Decoding apparatus comprising a first Max-Log-MAPdecoder, a first weight applicator, a second Max-Log-MAP decoder, and asecond weight applicator, the decoder performing processing iterations,the first Max-Log-MAP decoder providing extrinsic information includinga systematic error due to the Max-Log approximation, the first weightapplicator being connected to the first Max-Log-MAP decoder so as toapply weights to the extrinsic information from the first decoder tocompensate for the systematic error due to the Max-Log approximation andbeing connected to the second Max-Log-MAP decoder so as to provide theweighted extrinsic information as a priori information to the secondMax-Log-MAP decoder, and the second Max-Log-MAP decoder providingextrinsic information including a systematic error due to the Max-Logapproximation, the second weight applicator being connected to the firstMax-Log-MAP decoder so as to apply weights to the extrinsic informationfrom the second decoder to compensate for the systematic error due tothe Max-Log approximation and being connected to the first Max-Log-MAPdecoder so as to provide the weighted extrinsic information as a prioriinformation to the first Max-Log-MAP decoder for the next processingiteration.
 7. Decoding apparatus according to claim 6, in which, of theweights, a weight (w_(a) ^(i)) is applied to a priori information(Λ_(a)(x_(t,0))) which is in the form of a Log Likelihood Ratio about aparity bit, and a weight (w_(c) ^(i)) is applied to a priori information(Λ_(c)(x_(t,0))) which is in the form of a Log Likelihood Ratio about asystematic bit.
 8. Decoding apparatus according to claim 6, in whichvalues of the weights (w_(a) ^(i), w_(c) ^(i)) to be applied relate tooptimum weights, the optimum weights being optimum in maximising thetransfer of mutual information from one Max-log-MAP decoding operationto the next.
 9. Decoding apparatus according to claim 8, in which theoptimum weights are normalised to provide the weights to be applied soas to maintain a natural ratio between Λ_(c)(x_(t,0)) and Λ_(c)(x_(t,1))undisturbed.
 10. Decoding apparatus according to claim 6, in which theweights to be applied are computed once and used repeatedly thereafter.